Funny you should raise this - I have a similar issue for a viewpoint character, and I've just been working on editing the sequence. He is told to adopt a new name, Oswald, instead of his old one, Walt. In the chapter where this happens I continue to refer to him as Walt. Only in the next chapter do I call him Oswald throughout.
I don't have a line like yours, though. It feels a bit on the nose, and I'm sure the reader gets what's going on. There are subtler ways of reminding them that Oswald is Walt, like having him recall things that happened to him earlier in the narrative, or remember his backstory that readers already know about.
Only when he has a flashback to a traumatic incident does he become Walt again in the narrative, reflecting what's going on in his head. It might be confusing, but again I think readers will get why it happens. Ideally they won't even notice, but the emotional impact of his past coming back will be felt through the use of his real name.
I'm not saying there's a right and a wrong way. I guess the two important factors are: what's the effect on the reader - confusion bad, identification good; and does your decision reflect how the character thinks about himself? Walt is happy to be Oswald, because he's running away from Walt's past, and because his life is at stake. Also, he's young, unsure of himself, and very much in awe of / controlled by the person who changes his name.
If your Danny is adopting Frank as a very knowing disguise, but continues to think of himself as Danny, you might keep referring to him as Danny. Doing so will let the reader know that he's maintaining his own identity, at least to himself. Calling him Frank in your narration, conveys that he is totally immersed in his new identity, and it's not just a superficial thing. (And then it might be worth a paragraph exploring his thinking.)
Good luck with it!